change particle color on interval
If you want the change to be frame dependent, as opposed to lifetime dependent, I would suggest using a Value node with #frame written as it's input (makes value = current frame at any point), and then using some math to get the correct interval for your needs. Use this as a material for the object you're instancing as a particle.
In the example below, I took the time that the Sine of the frame number is greater than 0.5 (half the time) and used that as a mix factor for a MixRGB with the two colors. This gives particles that alternate between red and green every other frame. You can also use a ColorRamp if you want more control/multiple colors.

Lastly, this works in both Cycles and EEVEE.